WAEC - Further Mathematics (2014 - No. 22)
Integrate \((x - \frac{1}{x})^{2}\) with respect to x.
\(\frac{1}{3}(x - \frac{1}{x})^{3} + c\)
\(\frac{x^{3}}{3} - x\sqrt{\frac{1}{x^{3}}} + c\)
\(\frac{x^{3}}{3} - 2x + \frac{1}{x^{3}} + c\)
\(\frac{x^3}{3} - 2x - \frac{1}{x} + c\)
Explanation
\((x - \frac{1}{x})^{2} = x^2 - 2 + \frac{1}{x^2}\)
\(\int (x^2 + \frac{1}{x^2} - 2) \mathrm {d} x\)
= \(\int (x^2 + x^{-2} - 2) \mathrm {d} x\)
= \(\frac{x^3}{3} - 2x - \frac{1}{x}\)
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